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Evaluate the integral below:

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\$$\\int\\int_R\\var{k2}(x^2+y^2)dxdy\$$

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where \$$R\$$ is the region of the plane enclosed by \$$\\var{a1}\\le \\var{k}xy\\le \\var{b1}\$$   and   \$$\\var{a2}\\le x^2-y^2\\le \\var{b2}\$$.

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\$$\\int\\int_R\\var{k2}(x^2+y^2)dxdy\$$

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\$$R\$$ is the region of the plane enclosed by \$$\\var{a1}\\le \\var{k}xy\\le \\var{b1}\$$   and   \$$\\var{a2}\\le x^2-y^2\\le \\var{b2}\$$

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Let \$$u=\\var{k}xy\$$  and let  \$$v=x^2-y^2\$$

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Limits:

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\$$\\var{a1}\\le u\\le \\var{b1}\$$  and  \$$\\var{a2}\\le v\\le \\var{b2}\$$

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Jacobian:

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\$$\\begin{vmatrix} \\frac{du}{dx}&\\frac{du}{dy}\\\\ \\frac{dv}{dx}&\\frac{dv}{dy}\\\\ \\end{vmatrix}=\\begin{vmatrix} \\var{k}y&\\var{k}x\\\\2x&-2y\\\\\\end{vmatrix}=\\simplify{-2*{k}}y^2-\\simplify{2*{k}}x^2=-\\simplify{2*{k}}(x^2+y^2)\$$

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\$$The\\,Jacobian\\,is\\,the\\,absolute\\,value\\,of\\,the\\,determinant:J=\\simplify{2*{k}}(x^2+y^2)\$$

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\$$dxdy=\\frac{1}{\\simplify{2*{k}}(x^2+y^2)}dudv\$$

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\$$\\implies \\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\var{k2}(x^2+y^2)\\frac{1}{\\simplify{2*{k}}(x^2+y^2)}dudv=\\int_\\var{a2}^\\var{b2}\\int_\\var{a1}^\\var{b1}\\frac{\\var{k2}}{\\simplify{2*{k}}}dudv\$$

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Inner integral:

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\$$\\int_\\var{a1}^\\var{b1}\\frac{\\var{k2}}{\\simplify{2*{k}}}du\$$

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\$$=\\frac{\\var{k2}}{\\simplify{2*{k}}}u\\big|_\\var{a1}^\\var{b1}\$$

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\$$=\\frac{\\var{k2}}{\\simplify{2*{k}}}(\\var{b1})-\\frac{\\var{k2}}{\\simplify{2*{k}}}(\\var{a1})\$$

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\$$=\\var{inner}\$$

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Outer integral:

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\$$\\int_\\var{a2}^\\var{b2}\\var{inner}dv\$$

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\$$=\\var{inner}v\\big|_\\var{a2}^\\var{b2}\$$

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\$$=\\var{inner}(\\var{b2})-\\var{inner}(\\var{a2})\$$

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\$$=\\simplify{{inner}*({b2}-{a2})}\$$

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